# Secret Messages and Matrices

Carol West (cwwest@dimacs.rutgers.edu)

Lesson Description
This lesson uses matrices and linear equations to decrypt a secret message. Each student receives a part of the encoded message to decode by means of inverse matrices.

Prerequisites
Multiplying matrices
Writing a system of linear equations in matriz form
Finding the inverse of a 2x2 matrix
Solving a system of 2 linear equations using iverse matrices

Preparation
Begin with matching the letters of the alphabet to the numbers 1-26 (A=1, B=2,...) Ask "What if you have 27?...31?...40?...53? to give the idea of hte circular action of mod 26 arithmetic.

The the teacher will write a secret message on the board or overhead, along with a 2x2 matrix representing a system of linear equations. Each student receives a slip of paper with two numbers on it, and a position number in the message where their decrypted letters will fit. Attached are two equations and a sheet containing pairs of numbers, along with a secret message encryption. Each position contains 2 letters of the message.

Activity Description
On the board are spaces marked with position numbers for pairs of letters in the message, and a matrix representing a system of equations. This matrix is called the "key" and without it, the message cannot be decoded easily. Students are given a slip of paper with a position number and two other numbers, a value for f and a value for g. Note that at least 2 students obtain the same numbers for this particular message, which has only 12 positions. Students are told that the spaces on the board are for the secret message which they are to decode using inverse matrices. Each student knows a piece of the message, but alone, cannot determine the whole message.

Explain the A=1, B=2, ... or let them write out a chart on the board. If the number is larger than 26, use the value mod26.

Teacher Notes
It is a good idea to require each student to turn in the solution of their part of the message, since more than one student has the same pair, in order to check that each student has actually done their own work.

The students in my Honors Algebra 2 Trig class enjoyed moving around the room and sharing their solutions with each other, then filling in the letters and guessing the message even when some of the letters were still missing.

Their homework was to make up and encode a short message of their own to share in class, and to write a critique about the activity, commenting on how well they understood it, and what could be done to improve the lesson.